I'm attaching a link to an inro that had a ojime that looked very much like mine. Yours and the ones shown by Robert Liu looked more red than mine, but the one on the link is very similar in color. Scroll down about halfway down the page, the title for the item is: Tsuishu laquer Inro with sages and netsuke.

The second link is to a discussion on the International Netsuke Society Forum which discusses this type of ojime.

It looks like the redness is probably an applied dye and that mine has little or none of the dye.

I'm indebted to you for all the information and time you are providing, I appreciate it so much!